## Introduction

For highly accurate quantum chemical calculations, the effects of relativity can hardly be ignored. Relativistic effects may be crucial to explain the reaction dynamics when heavy elements, of special interest in catalysis, are involved, even at the Hartree–Fock or density functional theory (DFT) level. The zeroth-order relativistic approximation (ZORA) 1, 2 was rediscovered during the 1990s 3–5 and has recently again experienced a surge of interest 6, 7. This approach in its simplest scalar form reduces the Dirac equation 8 to a one-component equation in which the kinetic energy operator is replaced by a potential dependent operator:

Matrix elements of the kinetic energy operator thus become 9:

where *V* is the Coulomb potential due to electrons and nuclei. Within the context of DFT theory, the inverse operator is generally determined using numerical integration. In our approach we used a resolution of the identity operator employing a suitably chosen orthogonalized internal basis and a matrix inversion 10. To recover the correct relativistic limit, the kinetic energy operator is split of

A major problem plaguing this operator is the fact that the presence of the potential in the inverse coulomb operator causes a lack of gauge invariance, i.e., if a constant is added to the potential this constant is not properly reproduced in the total energy. In our previous paper 11, we proposed to calculate the ZORA corrections separately for the atoms. To eliminate all dependence on the atomic coordinates both the potential and all matrix elements are calculated just for the atoms. Thus, we use

The relativistic corrections to the kinetic energy matrix elements may thus be calculated separately for each unique atom or even stored, requiring negligible computing time. Subsequently, they are only added to the kinetic energy integrals for basis functions that are both centered on the atom concerned. This way the influence of the potential of other atoms is completely eliminated, curing the gauge problem that results from this potential. Also, as the ZORA corrections move with the atoms, they have no other effect on the gradients than the change in the wave function due to the different kinetic energy operator. Thus, no special gradient-related terms need to be programmed. The reasoning that no extra terms enter our gradient expressions can be applied to the second derivatives as well. As the ZORA corrections move with the atoms no second derivatives of these corrections are required. The approach proved to be very accurate 11, as the elimination of the Gauge dependence well outweighs the minor effect of the other atoms on the atomic potential.

Other methods have been proposed to facilitate the calculation of gradients within the ZORA formalism. They include using a model potential 5, to allow analytic expressions for the gradients and applying corrections to the potentials to approximately eliminate the influence of the other atoms 6, 12. None of these methods, including our atomic ZORA, is gauge independent in a general sense. An extension of our atomic idea to the case of polarizabilities, for example, would entail simply ignoring the electric field in the evaluation of the ZORA correction.